The variance-ratio (VR) test statistic, which is based on k-period differences of the data, is commonly used in empirical finance and economics to test the random walk hypothesis. We obtain the asymptotic power function of the VR test statistic when the differencing period k is increasing with the sample size n such that k/n → δ > 0. We show that the test is inconsistent against a variety of mean-reverting alternatives, confirm the result in simulations, and then characterize the functional form of the asymptotic power in terms of δ and these alternatives.

This paper studies power functions and envelopes for covariate augmented unit root tests. The power functions are calculated by integrating the characteristic function, allowing accurate evaluation of the power envelope and the power functions. Using the power functions, we study the selection among point optimal invariant unit root tests. An “optimal” point optimal test is proposed based on minimizing the integrated power difference. We find that when there are covariate effects, optimal tests use a local alternative where the power envelope has an approximate value of 0.75.We thank Pentti Saikkonen and two referees for helpful comments.

In this paper, two competing types of multistep predictors, i.e., plug-in and direct predictors, are considered in autoregressive (AR) processes. When a working model AR(k) is used for the h-step prediction with h > 1, the plug-in predictor is obtained from repeatedly using the fitted (by least squares) AR(k) model with an unknown future value replaced by their own forecasts, and the direct predictor is obtained by estimating the h-step prediction model's coefficients directly by linear least squares. Under rather mild conditions, asymptotic expressions for the mean-squared prediction errors (MSPEs) of these two predictors are obtained in stationary cases. In addition, we also extend these results to models with deterministic time trends. Based on these expressions, performances of the plug-in and direct predictors are compared. Finally, two examples are given to illustrate that some stationary case results on these MSPEs can not be generalized to the nonstationary case.The author is deeply grateful to the co-editor Pentti Saikkonen and two referees for their helpful suggestions and comments on a previous version of this paper.

This paper investigates the asymptotic theory for a vector autoregressive moving average–generalized autoregressive conditional heteroskedasticity (ARMA-GARCH) model. The conditions for the strict stationarity, the ergodicity, and the higher order moments of the model are established. Consistency of the quasi-maximum-likelihood estimator (QMLE) is proved under only the second-order moment condition. This consistency result is new, even for the univariate autoregressive conditional heteroskedasticity (ARCH) and GARCH models. Moreover, the asymptotic normality of the QMLE for the vector ARCH model is obtained under only the second-order moment of the unconditional errors and the finite fourth-order moment of the conditional errors. Under additional moment conditions, the asymptotic normality of the QMLE is also obtained for the vector ARMA-ARCH and ARMA-GARCH models and also a consistent estimator of the asymptotic covariance.The authors thank the co-Editor, Bruce Hansen, and two referees for very helpful comments and suggestions and acknowledge the financial support of the Australian Research Council.

This paper analyzes the large sample behavior of the seasonal unit root tests of Dickey, Hasza, and Fuller (1984, Journal of the American Statistical Association 79, 355–367) when applied to a series that admits a unit root at the zero but not seasonal spectral frequencies. We show that in such cases the Dickey et al. statistics have nondegenerate limiting distributions. Consequently, there is a nonzero probability that, taken in isolation, they will lead the applied researcher to accept the seasonal unit root null hypothesis and hence, incorrectly, take seasonal differences of the series, even asymptotically. The same conclusion holds if the process displays unit root behavior at any of the zero and/or seasonal frequencies. Our results therefore prove a conjecture made on the basis of Monte Carlo simulation evidence, in Ghysels, Lee, and Noh (1994, Journal of Econometrics 62, 415–442) that the tests of Dickey et al., unlike those of Hylleberg, Engle, Granger, and Yoo (1990, Journal of Econometrics 44, 215–238), are unable to separate between unit roots at the zero and seasonal frequencies.I thank Peter Burridge, Bruce Hansen, and three anonymous referees for helpful comments on earlier drafts of this paper.

This paper proposes a test of the null of underidentification in the nonlinear-in-parameters generalized method of moments model. It can be thought of as a nonlinear analog of the usual linear instrumental variables first-stage F-test. It can be used as a diagnostic to warn a researcher when conventional asymptotic theory is unlikely to work well.I am grateful to Don Andrews, Jon Faust, John Fernald, Jim Stock, and two anonymous referees for their helpful comments on earlier drafts of this manuscript. I am also grateful to George Tauchen for providing me with the code for generating artificial asset price data. All errors are my sole responsibility.

It was June 1940. The Second World War was already raging in full swing in its devastation. A young man not yet 20 set out on a 500-mile train journey from the coastal city of Vishakapatnam, India, to Calcutta, the second largest city of the British Empire, after obtaining a first-class first degree in mathematics and with a glimmer of hope of finding a job in the military. The young man was not so lucky; he was deemed too young for the job. However, while in Calcutta, through a chance encounter, he visited the Indian Statistical Institute (ISI) founded in 1931 by Professor Prasanta Chandra Mahalanobis, a Cambridge-trained physicist. As a last resort he applied for a one-year training program in statistics there. Very promptly he received a positive reply from Professor Mahalanobis admitting him to the Training Section of the ISI from January 1, 1941. This young man was none other than perhaps the most well known statistician in the world today, Calyampudi Radhakrishna Rao (or simply Dr. Rao, as he was universally known to his students and colleagues at the ISI. The “professor” title was reserved for Mahalanobis). Such was the beginning of his spectacular career in statistics, a field chosen as a last resort, and he never looked back.

Over the past decade financial econometrics has evolved from a small subfield of econometrics to a rapidly growing area of research that maintains its own journals and dominates the applied time series sessions in many major conferences. Financial econometrics is a conglomerate of statistical models and techniques to summarize regular features of financial data sets and to evaluate theoretical models and hypotheses that have been set forth by financial economists. Besides being a focal point of intellectual curiosity, financial econometrics also provides a rich tool kit that is used by practitioners around the world to forecast volatilities of asset return, optimize portfolio allocations, price derivatives, and manage market risk. Courses in financial econometrics are being offered at all major universities and business schools at the master's and doctoral levels. Given the importance and popularity of the field, it is not surprising that it has become the subject of textbooks and monographs, one of them being the new book Financial Econometrics by Christian Gourieroux and Joann Jasiak.This review has benefited from discussions with Frank Diebold and participants of the Penn Econometrics Lunch.

Fixed effects estimation of the population-averaged slopes in a panel data random coefficient model.

ARMA representation of squared Markov switching heteroskedastic models—solution.